Relativity & Distance-Time: Is It True?

[Meanwhile]

Is it correct that the time it takes to travel a distance from the point of view of the traveler (measured by their clock) is the same as it is in classical mechanics?

I.e. if you start accelerating at a constant rate, at some point you will go "faster than light" if you define speed as distance divided by onboard time (despite you'll always stay sub c for an external observer)?

 

[Nugatory]

Is it correct that the time it takes to travel a distance from the point of view of the traveler (measured by their clock) is the same as it is in classical mechanics?

Yes. If the distance traveled is $D$, the time taken on the journey is $T$, and the speed is $S$ and all are measured by the same inertial observer, then $D=ST$, $T=D/S$, $S=D/T$. But you have to remember that bit about "all are measured by the same inertial observer" - you cannot mix the values seen by different inertial observers or measured in different reference frames in these formulas.

Thus, it does not follow that:

I.e. if you start accelerating at a constant rate, at some point you will go "faster than light" if you define speed as distance divided by onboard time (despite you'll always stay sub c for an external observer)?

We know what your onboard time is, but what value do we use for the distance? As the ship accelerates, its speed increases so the distance between origin and destination as measured by the onboard observer is being reduced by length contraction. There is no moment in the journey, including at the end when the traveller zooms past the destination at very close to $c$, that the distance as measured by the traveler divided by the travel time comes out greater than $c$.

 

[A.T.]

I.e. if you start accelerating at a constant rate, at some point you will go "faster than light" if you define speed as distance divided by onboard time (despite you'll always stay sub c for an external observer)?

See:
http://en.wikipedia.org/wiki/Proper_velocity

 

[Meanwhile]

Thank you. I was always wandering why such an obvious concept is hidden so deep in the theory. I think relativity should be taught "backwards": proper velocity first, THEN time dilation and so on.

 

[sardar]

The concept of relativity, as introduced by Albert Einstein, states that the laws of physics are the same for all observers in uniform motion. This means that the measurement of time and distance can vary depending on the observer's frame of reference. In classical mechanics, time and distance are absolute and do not change based on the observer's perspective.

In the context of relativity, the statement that the time it takes to travel a distance is the same for the traveler (measured by their clock) as it is in classical mechanics is not entirely accurate. This is because in relativity, time and distance are not absolute and can vary depending on the observer's frame of reference. For example, in the theory of special relativity, time dilation occurs when an object moves at high speeds, causing time to pass slower for that object compared to a stationary observer.

The idea that an object can accelerate at a constant rate and eventually reach a speed faster than light is also not consistent with the principles of relativity. According to the theory of special relativity, the speed of light is the maximum speed at which any object can travel, and it cannot be exceeded. This is why the concept of time dilation occurs, as an object approaching the speed of light would experience time at a slower rate.

In conclusion, the concepts of relativity and classical mechanics are fundamentally different, and the idea that the time it takes to travel a distance is the same for the traveler in both cases is not entirely correct. Relativity introduces the idea that time and distance are relative and can vary depending on the observer's frame of reference, while classical mechanics assumes them to be absolute.